In neural networks, what does the term depth generally mean?
Very good question, I have heard it refered to all the 3 ways you described. I am not sure if there is a purely objective answer so I will answer with what I understand by "depth" generally:


If speaking of model architecture: the number of layers of the model.
If speaking of CNN layers: the number of filters (not to be confused with the size of the kernel).
If speaking of RNN layers: the number of time steps (temporal "depth").


On the other hand I have never heard about depth regarding to FC layers. As I said the term "depth" is very subjective depending on the speakers. It should not be this way, and maybe theoretically is not, but I speak from what I heard in my experience.